A $20$kW$,4-$pole induction motor is designed to operate from a $440$V$,50$Hz$,$ three$-$phasesupply, and when operating at full power on this supply it runs at $1470$rpm$.$ The motorefficiency is $90\%$ under both conditions.

$($i$)$ What supply frequency will be needed to make this motor run at $1270$rpm whiledelivering a shaft power of $12\mathrm{.}5$kW$.$

$($ii$)$ If the motor were supplied from a sinusoidal variable frequency source, whatvoltage and current will need to be supplied to it when running at $1365$rpm at$12\mathrm{.}5$kW if the power factor of the motor is $0\mathrm{.}85?$